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Unformatted text preview: he diagonal
Proof. If B = A∗ A, apply QR Factorization to A:
B = A∗ A = (QR)∗ (QR) = R∗ Q∗ QR = R∗ IR = R∗ R
Letting L = R∗ yields the desired result.
We can use the QR Factorization to get a triangularization algorithm. However, we note A = QR
is asymmetric, whereas Schur wants A = U ∗ T U .
2.5.9 Proposition. Let A ∈ Mn . Choose A0 = A = Q0 R0 . Next, let An+1 = Rn Qn =
Qn+1 Rn+1 , with Qn , Rn chosen as described above for any n ∈ N, then each An is unitarily
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